Material property measurements using multiple frequency atomic force microscopy

ABSTRACT

Apparatus and techniques for extracting information carried in higher eigenmodes or harmonics of an oscillating cantilever or other oscillating sensors in atomic force microscopy and related MEMs work are described. Similar apparatus and techniques for extracting information using contact resonance with multiple excitation signals are also described.

CROSS-REFERENCE

The present application is a divisional application of U.S. Ser. No.14/052,750, filed Oct. 13, 2013; which was a continuation of Ser. No.13/241,689 filed Sep. 23, 2011, now U.S. Pat. No. 8,555,711 issued Oct.15, 2013, which is a continuation of U.S. Ser. No. 12/214,031 filed Jun.16, 2008, now U.S. Pat. No. 8,024,963 issued Sep. 27, 2011, which is acontinuation-in-part to U.S. application Ser. No. 11/544,130 filed Oct.5, 2006, now U.S. Pat. No. 7,603,891 issued Oct. 20, 2009 which claimspriority from provisional application No. 60/936,147 filed Jun. 18,2007, the disclosures of which are hereby incorporated by reference.

BACKGROUND OF THE INVENTION

For the sake of convenience, the current description focuses on systemsand techniques that may be realized in a particular embodiment ofcantilever-based instruments, the atomic force microscope (AFM).Cantilever-based instruments include such instruments as AFMs, molecularforce probe instruments (1D or 3D), high-resolution profilometers(including mechanical stylus profilometers), surface modificationinstruments, chemical or biological sensing probes, and micro-actuateddevices. The systems and techniques described herein may be realized insuch other cantilever-based instruments.

An AFM is a device used to produce images of surface topography (and/orother sample characteristics) based on information obtained fromscanning (e.g., rastering) a sharp probe on the end of a cantileverrelative to the surface of the sample. Topographical and/or otherfeatures of the surface are detected by detecting changes in deflectionand/or oscillation characteristics of the cantilever (e.g., by detectingsmall changes in deflection, phase, frequency, etc., and using feedbackto return the system to a reference state). By scanning the proberelative to the sample, a “map” of the sample topography or other samplecharacteristics may be obtained.

Changes in deflection or in oscillation of the cantilever are typicallydetected by an optical lever arrangement whereby a light beam isdirected onto the cantilever in the same reference frame as the opticallever. The beam reflected from the cantilever illuminates a positionsensitive detector (PSD). As the deflection or oscillation of thecantilever changes, the position of the reflected spot on the PSDchanges, causing a change in the output from the PSD. Changes in thedeflection or oscillation of the cantilever are typically made totrigger a change in the vertical position of the cantilever baserelative to the sample (referred to herein as a change in the Zposition, where Z is generally orthogonal to the XY plane defined by thesample), in order to maintain the deflection or oscillation at aconstant pre-set value. It is this feedback that is typically used togenerate an AFM image.

AFMs can be operated in a number of different sample characterizationmodes, including contact mode where the tip of the cantilever is inconstant contact with the sample surface, and AC modes where the tipmakes no contact or only intermittent contact with the surface.

Actuators are commonly used in AFMs, for example to raster the probe orto change the position of the cantilever base relative to the samplesurface. The purpose of actuators is to provide relative movementbetween different parts of the AFM; for example, between the probe andthe sample. For different purposes and different results, it may beuseful to actuate the sample, the cantilever or the tip or somecombination of both. Sensors are also commonly used in AFMs. They areused to detect movement, position, or other attributes of variouscomponents of the AFM, including movement created by actuators.

For the purposes of the specification, unless otherwise specified, theterm “actuator” refers to a broad array of devices that convert inputsignals into physical motion, including piezo activated flexures, piezotubes, piezo stacks, blocks, bimorphs, unimorphs, linear motors,electrostrictive actuators, electrostatic motors, capacitive motors,voice coil actuators and magnetostrictive actuators, and the term“position sensor” or “sensor” refers to a device that converts aphysical parameter such as displacement, velocity or acceleration intoone or more signals such as an electrical signal, including capacitivesensors, inductive sensors (including eddy current sensors),differential transformers (such as described in co-pending applicationsUS20020175677A1 and US20040075428A1, Linear Variable DifferentialTransformers for High Precision Position Measurements, andUS20040056653A1, Linear Variable Differential Transformer with DigitalElectronics, which are hereby incorporated by reference in theirentirety), variable reluctance, optical interferometry, opticaldeflection detectors (including those referred to above as a PSD andthose described in co-pending applications US20030209060A1 andUS20040079142A1, Apparatus and Method for Isolating and MeasuringMovement in Metrology Apparatus, which are hereby incorporated byreference in their entirety), strain gages, piezo sensors,magnetostrictive and electrostrictive sensors.

In both the contact and AC sample-characterization modes, theinteraction between the stylus and the sample surface induces adiscernable effect on a probe-based operational parameter, such as thecantilever deflection, the cantilever oscillation amplitude, the phaseof the cantilever oscillation relative to the drive signal driving theoscillation or the frequency of the cantilever oscillation, all of whichare detectable by a sensor. In this regard, the resultantsensor-generated signal is used as a feedback control signal for the Zactuator to maintain a designated probe-based operational parameterconstant.

In contact mode, the designated parameter may be cantilever deflection.In AC modes, the designated parameter may be oscillation amplitude,phase or frequency. The feedback signal also provides a measurement ofthe sample characteristic of interest. For example, when the designatedparameter in an AC mode is oscillation amplitude, the feedback signalmay be used to maintain the amplitude of cantilever oscillation constantto measure changes in the height of the sample surface or other samplecharacteristics.

The periodic interactions between the tip and sample in AC modes inducescantilever flexural motion at higher frequencies. Measuring the motionallows interactions between the tip and sample to be explored. A varietyof tip and sample mechanical properties including conservative anddissipative interactions may be explored. Stark, et al., have pioneeredanalyzing the flexural response of a cantilever at higher frequencies asnonlinear interactions between the tip and the sample. In theirexperiments, they explored the amplitude and phase at numerous higheroscillation frequencies and related these signals to the mechanicalproperties of the sample.

Unlike the plucked guitar strings of elementary physics classes,cantilevers normally do not have higher oscillation frequencies thatfall on harmonics of the fundamental frequency. The first three modes ofa simple diving board cantilever, for example, are at the fundamentalresonant frequency (f₀), 6.19 f₀ and 17.5 f₀. An introductory text incantilever mechanics such as Sarid has many more details. Throughcareful engineering of cantilever mass distributions, Sahin, et al.,have developed a class of cantilevers whose higher modes do fall onhigher harmonics of the fundamental resonant frequency. By doing this,they have observed that cantilevers driven at the fundamental exhibitenhanced contrast, based on their simulations on mechanical propertiesof the sample surface. This approach is has the disadvantage ofrequiring costly and difficult to manufacture special cantilevers.

The simple harmonic oscillator (SHO) model gives a convenientdescription at the limit of the steady state amplitude of the eigenmodeA of a cantilever oscillating in an AC mode:

$\begin{matrix}{{A = \frac{F_{0}\text{/}m}{\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2} - \left( {{\omega\omega}_{0}\text{/}Q} \right)^{2}}}},} & {SHOAmp0}\end{matrix}$where F₀ is the drive amplitude (typically at the base of thecantilever), m is the mass, ω is the drive frequency in units ofrad/sec, ω₀ is the resonant frequency And Q is the “quality” factor, ameasure of the damping.

If, as is often the case, the cantilever is driven through excitationsat its base, the expression becomes

$\begin{matrix}{{A = \frac{A_{drive}\omega_{0}^{2}}{\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2} - \left( {\omega_{0}\omega\text{/}Q} \right)^{2}}}},} & {SHOAmp1}\end{matrix}$

where F₀/m has been replaced with A_(drive)ω₀ ², where A_(drive) is thedrive amplitude (at the oscillator).

The phase angle φ is described by an associated equation

$\begin{matrix}{\varphi = {{\tan^{- 1}\left\lbrack \frac{{\omega\omega}_{0}}{Q\left( {\omega_{0}^{2} - \omega^{2}} \right)} \right\rbrack}.}} & {SHOPhase}\end{matrix}$

When these equations are fulfilled, the amplitude and phase of thecantilever are completely determined by the user's choice of the drivefrequency and three independent parameters: A_(drive), ω₀ and Q.

In some very early work, Martin, et al., drove the cantilever at twofrequencies. The cantilever response at the lower, non-resonantfrequency was used as a feedback signal to control the surface trackingand produced a topographic image of the surface. The response at thehigher frequency was used to characterize what the authors interpretedas differences in the non-contact forces above the Si and photo-resiston a patterned sample.

Recently, Rodriguez and Garcia published a theoretical simulation of anon-contact, attractive mode technique where the cantilever was drivenat its two lowest Eigen frequencies. In their simulations, they observedthat the phase of the second mode had a strong dependence on the Hamakerconstant of the material being imaged, implying that this techniquecould be used to extract chemical information about the surfaces beingimaged. Crittenden et al. have explored using higher harmonics forsimilar purposes.

There are a number of modes where the instrument is operated in a hybridmode where a contact mode feedback loop is maintained while someparameter is modulated. Examples include force modulation andpiezo-response imaging.

Force modulation involves maintaining a contact mode feedback loop whilealso driving the cantilever at a frequency and then measuring itsresponse. When the cantilever makes contact with the surface of thesample while being so driven, its resonant behavior changessignificantly. The resonant frequency typically increases, depending onthe details of the contact mechanics. In any event, one may learn moreabout the surface properties because force modulation is sensitive tothe elastic response of the sample surface. In particular, dissipativeinteractions may be measured by measuring the phase of the cantileverresponse with respect to the drive.

A well-known shortcoming of force modulation and other contact modetechniques is that the while the contact forces may be controlled well,other factors affecting the measurement may render it ill-defined. Inparticular, the contact area of the tip with the sample, usuallyreferred to as contact stiffness, may vary greatly depending on tip andsample properties. This in turn means that the change in resonance whilemaintaining a contact mode feedback loop, which may be called thecontact resonance, is ill-defined. It varies depending on the contactstiffness. This problem has resulted in prior art techniques avoidingoperation at or near resonance.

SUMMARY OF THE INVENTION

Cantilevers are continuous flexural members with a continuum ofvibrational modes. The present invention describes different apparatusand methods for exciting the cantilever simultaneously at two or moredifferent frequencies and the useful information revealed in the imagesand measurements resulting from such methods. Often, these frequencieswill be at or near two or more of the cantilever vibrational eigenmodes

Past work with AC mode AFMs has been concerned with higher vibrationalmodes in the cantilever, with linear interactions between the tip andthe sample. The present invention, however, is centered aroundnon-linear interactions between the tip and sample that couple energybetween two or more different cantilever vibrational modes, usually keptseparate in the case of linear interactions. Observing the response ofthe cantilever at two or more different vibrational modes has someadvantages in the case of even purely linear interactions however. Forexample, if the cantilever is interacting with a sample that has somefrequency dependent property, this may show itself as a difference inthe mechanical response of the cantilever at the different vibrationalmodes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a preferred embodiment for probing multiple eigenmodes of acantilever.

FIG. 2 shows a preferred embodiment for exciting voltage-dependentmotion in the cantilever probe.

FIG. 3 shows a preferred embodiment for probing an active device.

FIG. 4 shows Phase and amplitude shifts of the fundamental eigenmodewith and without the second eigenmode being driven.

FIG. 5A-5E shows Images of collagen fibers taken with the preferredembodiment.

FIGS. 6 and 7 show Two dimensional histogram plots of the amplitude andphase for the first and second eigenmodes.

FIG. 8 shows a Preferred embodiment for probing an active sample incontact while measuring dynamic contact properties (Dual FrequencyResonance Tracking Piezo Force Microscopy (DFRT PFM)).

FIG. 9 shows Resonance peaks in sweep of applied potential from dc to 2MHz.

FIGS. 10A-10D and 11 show Images of a piezoelectric sample when thecantilever potential was driven at two different frequencies, oneslightly below and the other slightly above the same contact resonancefrequency.

FIG. 12 show Amplitude versus frequency and phase versus frequencycurves simultaneous measured at different frequencies.

FIG. 13 show Amplitude and phase curves changing in response to varyingtip-sample interactions being driven first at a single frequency andthen at two different frequencies.

FIG. 14 show Amplitude versus frequency sweeps around the secondresonance made while feeding back on the first mode amplitude.

FIG. 15-16 Amplitude versus frequency and phase versus frequency curvessimultaneous measured at different frequencies.

FIG. 17-19 show Images of a piezoelectric sample when the cantileverpotential was driven at two different frequencies, one slightly belowand the other slightly above the same contact resonance frequency.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 is a block diagram of a preferred embodiment of an apparatus forprobing multiple eigenmodes of a cantilever in accordance with thepresent invention. The sample 1010 is positioned below the cantileverprobe 1020. The chip 1030 of the cantilever probe 1020 is driven by amechanical actuator 1040, preferably a piezoelectric actuator, but othermethods to induce cantilever motion known to those versed in the artcould also be used. The motion of the cantilever probe 1020 relative tothe frame of the microscope 1050 is measured with a detector 1060, whichcould be an optical lever or another method known to those versed in theart. The cantilever chip 1030 is moved relative to the sample 1010 by ascanning apparatus 1070, preferably a piezo/flexure combination, butother methods known to those versed in the art could also be used.

The motion imparted to the cantilever chip 1030 by actuator 1040 iscontrolled by excitation electronics that include at least two frequencysynthesizers 1080 and 1090. There could be additional synthesizers ifmore than two cantilever eigenmodes are to be employed. The signals fromthese frequency synthesizers could be summed together by an analogcircuit element 1100 or, preferably, a digital circuit element thatperforms the same function. The two frequency synthesizers 1080 and 1090provide reference signals to lock-in amplifiers 1110 and 1120,respectively. In the case where more than two eigenmodes are to beemployed, the number of lock-in amplifiers will also be increased. Aswith other electronic components in this apparatus, the lock-inamplifiers 1110 and 1120 can be made with analog circuitry or withdigital circuitry or a hybrid of both. For a digital lock-in amplifier,one interesting and attractive feature is that the lock-in analysis canbe performed on the same data stream for both eigenmodes. This impliesthat the same position sensitive detector and analog to digitalconverter can be used to extract information at the two distincteigenmodes.

The lock-in amplifiers could also be replaced with rms measurementcircuitry where the rms amplitude of the cantilever oscillation is usedas a feedback signal.

There are a number of variations in the FIG. 1 apparatus that a personskilled in the art could use to extract information relative to thedifferent eigenmodes employed in the present invention. Preferably, adirect digital synthesizer (DDS) could be used to create sine and cosinequadrature pairs of oscillating voltages, each at a frequency matched tothe eigenmodes of the cantilever probe 1030 that are of interest. Thisimplementation also allows dc voltages to be applied, allowing methodssuch as scanning Kelvin probing or simultaneous current measurementsbetween the tip and the sample. The amplitude and phase of eacheigenmode can be measured and used in a feedback loop calculated by thecontroller 1130 or simply reported to the user interface 1140 where itis displayed, stored and/or processed further in an off-line manner.Instead of, or in addition to, the amplitude and phase of the cantilevermotion, the quadrature pairs, usually designated x and y, can becalculated and used in a manner similar to the amplitude and phase.

In one method of using the FIG. 1 apparatus, the cantilever is driven ator near two or more resonances by the single “shake” piezo 1040.Operating in a manner similar to AC mode where the cantilever amplitudeis maintained constant and used as a feedback signal, but employing theteachings of the present invention, there are now a number of choicesfor the feedback loop. Although the work here will focus on using theamplitude of the fundamental (A₀), we were able to successfully imageusing one of the higher mode amplitudes (A_(i)) as a feedback signal aswell as a sum of all the amplitudes A₀+A₁+ . . . . One can also chooseto exclude one or more modes from such a sum. So for example, wherethree modes are employed, the sum of the first and second could be usedto operate the feedback loop and the third could be used as a carryalong signal.

Because higher eigenmodes have a significantly higher dynamic stiffness,the energy of these modes can be much larger that that of lowereigenmodes.

The method may be used to operate the apparatus with one flexural modeexperiencing a net attractive force and the other a net repulsive force,as well as operating with each mode experiencing the same net sign offorce, attractive or repulsive. Using this method, with the cantileverexperiencing attractive and repulsive interactions in differenteigenmodes, may provide additional information about sample properties.

One preferred technique for using the aforesaid method is as follows.First, excite the probe tip at or near a resonant frequency of thecantilever keeping the tip sufficiently far from the sample surface thatit oscillates at the free amplitude A₁₀ unaffected by the proximity ofthe cantilever to the sample surface and without making contact with thesample surface. At this stage, the cantilever is not touching thesurface; it turns around before it interacts with significant repulsiveforces.

Second, reduce the relative distance in the Z direction between the baseof the cantilever and the sample surface so that the amplitude of theprobe tip A₁ is affected by the proximity of the sample surface withoutthe probe tip making contact with the sample surface. The phase p₁ willbe greater than p₁₀, the free first eigenmode phase. This amplitude ismaintained at an essentially constant value during scanning without theprobe tip making contact with the sample surface by setting up afeedback loop that controls the distance between the base of thecantilever and the sample surface.

Third, keeping the first eigenmode drive and surface controllingfeedback loop with the same values, excite a second eigenmode of thecantilever at an amplitude A₂. Increase A₂ until the second eigenmodephase p₂ shows that the cantilever eigenmode is interacting withpredominantly repulsive forces; that is, that p₂ is less than p₂₀, thefree second eigenmode phase. This second amplitude A₂ is not included inthe feedback loop and is allowed to freely roam over a large range ofvalues. In fact, it is typically better if variations in A₂ can be aslarge as possible, ranging from 0 to A₂₀, the free second eigenmodeamplitude.

Fourth, the feedback amplitude and phase, A₁ and p₁, respectively, aswell as the carry along second eigenmode amplitude and phase, A₂ and p₂,respectively, should be measured and displayed.

Alternatively, the drive amplitude and/or phase of the second frequencycan be continually adjusted to maintain the second amplitude and/orphase at an essentially constant value. In this case, it is useful todisplay and record the drive amplitude and/or frequency required tomaintain the second amplitude and/or phase at an essentially constantvalue.

A second preferred technique for using the aforesaid method follows thefirst two steps of first preferred technique just described and thencontinues with the following two steps:

Third, keeping the first eigenmode drive and surface controllingfeedback loop with the same values, excite a second eigenmode (orharmonic) of the cantilever at an amplitude A₂. Increase A₂ until thesecond eigenmode phase p₂ shows that the cantilever eigenmode isinteracting with predominantly repulsive forces; that is, that p₂ isless than p₂₀, the free second eigenmode phase. At this point, thesecond eigenmode amplitude A₂ should be adjusted so that the firsteigenmode phase p₁ becomes predominantly less than p₁₀, the free firsteigenmode phase. In this case, the adjustment of the second eigenmodeamplitude A₂ has induced the first eigenmode of the cantilever tointeract with the surface in a repulsive manner. As with the firstpreferred technique, the second eigenmode amplitude A₂ is not used inthe tip-surface distance feedback loop and should be allowed rangewidely over many values.

Fourth, the feedback amplitude and phase, A₁ and p₁, respectively, aswell as the carry along second eigenmode amplitude and phase, A₂ and p₂,respectively, should be measured and displayed.

Either of the preferred techniques just described could be performed ina second method of using the FIG. 1 apparatus where the phase of theoscillating cantilever is used in a feedback loop and the oscillationfrequency is varied to maintain phase essentially constant. In thiscase, it is preferable to use the oscillation frequency as an input intoa z-feedback loop that controls the cantilever-sample separation.

Relative changes in various parameters such as the amplitude and phaseor in-phase and quadrature components of the cantilever at thesedifferent frequencies could also be used to extract information aboutthe sample properties.

A third preferred technique for using the aforesaid method provides analternative for conventional operation in a repulsive mode, that iswhere the tip is experiencing a net repulsive force. The conventionalapproach for so operating would be to use a large amplitude incombination with a lower set point, and a cantilever with a very sharptip. Using this third preferred technique, however, the operator begins,just as with the first two techniques, by choosing an amplitude and setpoint for the fundamental eigenmode that is small enough to guaranteethat the cantilever is experiencing attractive forces, that is, that thecantilever is in non-contact mode. As noted before, this operationalmode can be identified by observing the phase of the cantileveroscillation. In the non-contact case, the phase shift is positive,implying that the resonant frequency has been lowered. With theseconditions on the first eigenmode, the second eigenmode excitation canbe introduced and the amplitude, drive frequency and, if applicable, setpoint chosen with the following considerations in mind:

-   1. Both eigenmodes are in the attractive mode, that is to say that    the phase shift of both modes is positive, implying both eigenmode    frequencies have been shifted negatively by the tip-sample    interactions. Generally, this requires a small amplitude for the    second eigenmode.-   2. The fundamental eigenmode remains attractive while the second    eigenmode is in a state where the tip-sample interactions cause it    to be in both the attractive and the repulsive modes as it is    positioned relative to the surface.-   3. The fundamental eigenmode is in an attractive mode and the second    eiegenmode is in a repulsive mode.-   4. In the absence of any second mode excitation, the first eigenmode    is interacting with the surface in the attractive mode. After the    second eigenmode is excited, the first eigenmode is in a repulsive    mode. This change is induced by the addition of the second eigenmode    energy. The second eigenmode is in a state where the tip-sample    interactions cause it to be attractive and/or repulsive.-   5. The first eigenmode is in a repulsive mode and the second mode is    in a repulsive mode.

The transition from attractive to repulsive mode in the first eigenmode,as induced by the second eigenmode excitation, is illustrated in FIG. 4,where the amplitude and phase of the first and second eigenmodes areplotted as a function of the distance between the base of the cantileverand the surface of the sample. The point where the tip begins tointeract significantly with the surface is indicated with a solid line4000. The fundamental amplitude 4010 of the cantilever decreases as thecantilever starts to interact with the surface, denoted by the solidline 4000. The associated phase 4020 shows a positive shift, consistentwith overall attractive interactions. For these curves, the secondeigenmode amplitude is zero and therefore not plotted in the Figure (andneither is phase, for the same reason). Next, the second eigenmode isexcited and the same curves are re-measured, together with the amplitudeand phase of the second eigenmode, 4030 and 4040. There is a notablechange in the fundamental eigenmode amplitude 4050 and more strikingly,the fundamental eigenmode phase 4060. The fundamental phase in factshows a brief positive excursion, but then transitions to a negativephase shift, indicating an overall repulsive interaction between the tipand sample. The free amplitude of the first eigenmode is identical inboth cases, the only difference in the measurement being the addition ofenergy exciting the higher oscillatory eigenmode. This excitation issufficient to drive the fundamental eigenmode into repulsiveinteractions with the sample surface. Furthermore, the phase curve ofthe second eigenmode indicates that it is also interacting overallrepulsively with the sample surface.

More complicated feedback schemes can also be envisioned. For example,one of the eigenmode signals can be used for topographical feedbackwhile the other signals could be used in other feedback loops. Anexample would be that A₁ is used to control the tip-sample separationwhile a separate feedback loop was used to keep A₂ at an essentiallyconstant value rather than allowing it to range freely over many values.A similar feedback loop could be used to keep the phase of the secondfrequency drive p₂ at a predetermined value with or without the feedbackloop on A₂ being implemented.

As another example of yet another type of feedback that could be used,Q-control can also be used in connection with any of the techniques forusing the aforesaid method. Using Q-control on any or all of theeigenmodes employed can enhance their sensitivity to the tip-sampleforces and therefore mechanical or other properties of the sample. Itcan also be used to change the response time of the eigenmodes employedwhich may be advantageous for more rapidly imaging a sample. Forexample, the value of Q for one eigenmode could be increased and thevalue for another decreased. This may enhance the result of mixedattractive/repulsive mode imaging because it is generally easier to keepone eigenmode interacting with the sample in repulsive mode with areduced Q-value or, conversely, in attractive mode with an enhancedQ-value. By reducing the Q-value of the lowest eigenmode and enhancingthe Q-value of the next eigenmode, it is possible to encourage the mixedmode operation of the cantilever; the zeroth eigenmode will be inrepulsive mode while the first eigenmode will more likely remain inattractive mode. Q-control can be implemented using analog, digital orhybrid analog-digital electronics. It can be accomplished using anintegrated control system such as that in the Asylum ResearchCorporation MFP-3D Controller or by after-market modules such as thenanoAnalytics Q-box.

In addition to driving the cantilever at or near more than oneeigenmode, it is possible to also excite the cantilever at or near oneor more harmonics and/or one or more eigenmodes. It has been known forsome time that nonlinear interactions between the tip and the sample cantransfer energy into cantilever harmonics. In some cases this energytransfer can be large but it is usually quite small, on the order of apercent of less of the energy in the eigenmode. Because of this, theamplitude of motion at a harmonic, even in the presence of significantnonlinear coupling is usually quite small. Using the methods describedhere, it is possible to enhance the contrast of these harmonics bydirectly driving the cantilever at the frequency of the harmonic. Tofurther enhance the contrast of this imaging technique it is useful toadjust the phase of the higher frequency drive relative to that of thelower. This method improves the contrast of both conventionalcantilevers and the specially engineered “harmonic” cantileversdescribed by Sahin et al and other researchers.

On many samples, the results of imaging with the present invention aresimilar to, and in some cases superior to, the results of conventionalphase imaging. However, while phase imaging often requires a judiciouschoice of setpoint and drive amplitude to maximize the phase contrast,the method of the present invention exhibits high contrast over a muchwider range of imaging parameters. Moreover, the method also works influid and vacuum, as well as air and the higher flexural modes showunexpected and intriguing contrast in those environments, even onsamples such as DNA and cells that have been imaged numerous timesbefore using more conventional techniques.

Although there is a wide range of operating parameters that yieldinteresting and useful data, there are situations where more carefultuning of the operational parameters will yield enhanced results. Someof these are discussed below. Of particular interest can be regions inset point and drive amplitude space where there is a transition fromattractive to repulsive (or vice versa) interactions in one or more ofthe cantilever eigenmodes or harmonics.

The superior results of imaging with the present invention may be seenfrom an inspection of the images. An example is shown in FIG. 5. Forthis example, the FIG. 1 apparatus was operated using the fundamentaleigenmode amplitude as the error signal and the second eigenmode as acarry-along signal. The topography image 5010 in FIG. 5 shows collagenfibers on a glass surface, an image typical of results with conventionalAC mode from similar samples. The fundamental eigenmode amplitude image5020 is relatively similar, consistent with the fundamental eigenmodeamplitude being used in the feedback loop. The fundamental eigenmodephase channel image 5030 shows some contrast corresponding to edges inthe topography image. This is consistent with the interaction being moreattractive at these regions, again to be expected from surface energyconsiderations (larger areas in proximity will have larger long-rangeattractive forces). Since the fundamental eigenmode amplitude is beingheld relatively constant and there is a relationship between theamplitude and phase, the phase will be constrained, subject to energybalance and the feedback loop that is operating to keep the amplitudeconstant. The second eigenmode amplitude image 5040 shows contrast thatis similar to the fundamental eigenmode phase image 5030. However, thereare some differences, especially over regions thought to be contaminants5041 and 5042. Finally, the second eigenmode phase image 5050 shows themost surprisingly large amount of contrast. The background substrate5053 shows a bright, positive phase contrast. The putative contaminantpatches, 5041, 5042 and 5051 show strikingly dark, negative phase shiftcontrast. Finally, regions where the collagen fibers are suspended 5052show dark, negative phase contrast. In these last regions, the suspendedcollagen fibers are presumably absorbing some of the vibrational energyof the second eigenmode amplitude and thus, changing the response.

When an AFM is operated in conventional amplitude modulated (AM) AC modewith phase detection, the cantilever amplitude is maintained constantand used as a feedback signal. Accordingly, the values of the signalused in the loop are constrained not only by energy balance but also bythe feedback loop itself. Furthermore, if the amplitude of thecantilever is constrained, the phase will also be constrained, subjectto conditions discussed below. In conventional AC mode it is not unusualfor the amplitude to vary by a very small amount, depending on the gainsof the loop. This means that, even if there are mechanical properties ofthe sample that might lead to increased dissipation or a frequency shiftof the cantilever, the z-feedback loop in part corrects for thesechanges and thus in this sense, avoids presenting them to the user.

If the technique for using the present invention involves a mode that isexcited but not used in the feedback loop, there will be no explicitconstraints on the behavior of this mode. Instead it will range freelyover many values of the amplitude and phase, constrained only by energybalance. That is to say, the energy that is used to excite thecantilever motion must be balanced by the energy lost to the tip-sampleinteractions and the intrinsic dissipation of the cantilever. This mayexplain the enhanced contrast we observe in images generated with thetechniques of the present invention.

FIG. 6 demonstrates this idea more explicitly. The first image 6010 isan image of the number of pixels at different amplitudes (horizontalaxis) and phases (vertical axis) in the fundamental eigenmode data forthe collagen sample of FIG. 5. As expected, the amplitude values areconstrained to a narrow range around ˜0.6 Amax by the z-feedback loop.Constraining the amplitude values in turn, limits the values that thephase can take to the narrow range around 25°. Thus, when the pixelcounts are plotted, there is a bright spot 6020 with only smallvariations. Small variations in turn imply limited contrast. The secondimage 6030 plots the number of pixels at different amplitudes and phasesin the second eigenmode data for the collagen sample. Since theamplitude of this eigenmode was not constrained by a feedback loop, itvaries from ˜Amax to close to zero. Similarly, the phase ranges overmany values. This freedom allows greatly increased contrast in thesecond eigenmode images.

The present invention may also be used in apparatus that induce motionin the cantilever other than through a piezoelectric actuator. Thesecould include direct electric driving of the cantilever (“activecantilevers”), magnetic actuation schemes, ultrasonic excitations,scanning Kelvin probe and electrostatic actuation schemes.

Direct electric driving of the cantilever (“active cantilevers”) usingthe present invention has several advantages over conventional piezoforce microscopy (PFM) where the cantilever is generally scanned overthe sample in contact mode and the cantilever voltage is modulated in amanner to excite motion in the sample which in turn causes thecantilever to oscillate.

FIG. 2 is a block diagram of a preferred embodiment of an apparatus forusing the present invention with an active cantilever. This apparatushas similarities to that shown in FIG. 1, as well as differences. In theFIG. 2 apparatus, like the FIG. 1 apparatus, one of the frequencysources 1080 is used to excite motion of the cantilever probe 1020through a mechanical actuator 1040, preferably a piezoelectric actuator,but other methods to induce cantilever motion known to those versed inthe art could also be used, which drives the chip 1030 of the cantileverprobe 1020, However, in the FIG. 2 apparatus, the frequency source 1080communicates directly 2010 with the actuator 1040 instead of beingsummed together with the second frequency source 1090, as in the FIG. 1apparatus. The second frequency source 1090 in the FIG. 2 apparatus isused to vary the potential of the cantilever probe 1020 which in turncauses the sample 1010 to excite motion in the cantilever probe 1020 ata different eigenmode than that excited by the first frequency source1080. The resulting motion of the cantilever probe 1020 interacting withthe sample 1010 will contain information on the sample topography andother properties at the eigenmode excited by the first frequency source1080 and information regarding the voltage dependent properties of thesample at the eigenmode excited by the second frequency source 1090. Thesample holder 2030 can optionally be held at a potential, or at ground,to enhance the effect.

In one method of using the FIG. 2 apparatus, the amplitude of thecantilever at the frequency of the first source 1080 is used as theerror signal. The amplitude and phase (or in-phase and quadraturecomponents) at the frequency of the second source 1090 or a harmonicthereof will contain information about the motion of the sample andtherefore the voltage dependent properties of the sample. One example ofthese properties is the piezo-response of the sample. Another is theelectrical conductivity, charge or other properties that can result inlong range electrostatic forces between the tip and the sample.

FIG. 3 is a block diagram of a preferred embodiment of an apparatus forusing the present invention with the second frequency source modulatinga magnetic field that changes a property of the surface. In the FIG. 3apparatus, the situation with the first frequency source 1080 isidentical to the situation in the FIG. 2 apparatus. However, instead ofthe second frequency source 1090 being used to vary the potential of thecantilever probe 1020, as with the FIG. 2 apparatus, in the FIG. 3apparatus the second frequency source 1090 modulates the current throughan excitation coil 3010 which in turn modulates the magnetic state of amagnetic circuit element 3020. Magnetic circuit element 3020 could beused to modulate the field near an active sample or the excitation coil3010. Alternatively, magnetic circuit element 3020 could comprise thesample, as in the case of a magnetic recording head.

The FIG. 3 apparatus can be used with any other sort of ‘active’ samplewhere the interaction between the cantilever and the sample can bemodulated at or near one or more of the cantilever flexural resonancesby one of the frequency sources 1080 or 1090. This could also beextended to high frequency measurements such as described in Proksch etal., Appl. Phys. Lett., vol. (1999). Instead of the modulation describedin that paper, the envelope of the high frequency carrier could bedriven with a harmonic of one or more flexural resonances. This methodof measuring signals other than topographic has the advantage ofrequiring only one pass to complete as opposed to “LiftMode” or Nap modethat require temporally separated measurements of the topographic andother signals.

Another example of a preferred embodiment of an apparatus and method forusing the present invention is from the field of ultrasonic forcemicroscopy. In this embodiment, one or more eigenmodes are used for thez-feedback loop and one or more additional eigenmodes can be used tomeasure the high frequency properties of the sample. The high frequencycarrier is amplitude modulated and either used to drive the sampledirectly or to drive it using the cantilever as a waveguide. Thecantilever deflection provides a rectified measure of the sampleresponse at the carrier frequency.

Another group of embodiments for the present invention has similaritiesto the conventional force modulation technique described in theBackground to the Invention and conventional PFM where the cantilever isscanned over the sample in contact mode and a varying voltage is appliedto the cantilever. In general this group may be described as contactresonance embodiments. However, these embodiments, like the otherembodiments already described, make use of multiple excitation signals.

FIG. 8 is a block diagram of the first of these embodiments, which maybe referred to as Dual Frequency Resonance Tracking Piezo ForceMicroscopy (DFRT PFM). In the DFRT PFM apparatus of FIG. 8 thecantilever probe 1020 is positioned above a sample 1010 withpiezoelectric properties and scanned relative to the sample 1010 by ascanning apparatus 1070 using contact mode. Unlike conventional contactmode however the chip 1030 of the cantilever probe 1020, or thecantilever probe 1020 itself (alternative not shown), is driven byexcitation electronics that include at least two frequency synthesizers1080 and 1090. The cantilever probe 1020 responds to this excitation bybuckling up and down much as a plucked guitar string.

The signals from these frequency synthesizers could be summed togetherby an analog circuit element 1100 or, preferably, a digital circuitelement that performs the same function. The two frequency synthesizers1080 and 1090 provide reference signals to lock-in amplifiers 1110 and1120, respectively. The motion of the cantilever probe 1020 relative tothe frame of the microscope 1050 is measured with a detector 1060, whichcould be an optical lever or another method known to those versed in theart. The cantilever chip 1030 is moved relative to the sample 1010 inorder to maintain constant force by a scanning apparatus 1070,preferably a piezo/flexure combination, but other methods known to thoseversed in the art could also be used. The amplitude and phase of eachfrequency at which the cantilever probe 1020 is excited can be measuredand used in a feedback loop calculated by the controller 1130 or simplyreported to the user interface 1140 where it is displayed, stored and/orprocessed further in an off-line manner. Instead of, or in addition to,the amplitude and phase of the cantilever motion, the quadrature pairs,usually designated x and y, can be calculated and used in a mannersimilar to the amplitude and phase.

In one method of using the FIG. 8 apparatus, the topography of thesample would be measured in contact mode while the amplitude and phaseof the cantilever probe 1020 response to the applied potential at thelowest contact resonance and at the next highest contact resonance issimultaneously measured. The responses can be analyzed to determinewhether they originate from the actual piezoelectric response of thesample or from crosstalk between the topography and any electric forcesbetween the tip of the cantilever probe 1020 and the sample. Even moreinformation can be obtained if more frequencies are utilized.

FIG. 12 also shows three examples of the changes in the native phase12015 and amplitude 12010 of a cantilever with a resonant frequency f₀caused by interactions between the tip and the sample using DFRT PFMmethods. These examples are a subset of changes that can be observed. Inthe first example, the resonant frequency is significantly lowered tof₀′ but not damped. The phase 12085 and amplitude 12080 change butlittle relative to the native phase 12015 and amplitude 12010. In thesecond example the resonant frequency is again lowered to f₀′, this timewith damping of the amplitude. Here the phase 12095 is widened and theamplitude 12090 is appreciably flattened. Finally, in the third example,the resonant frequency is again dropped to f₀′, this time with areduction in the response amplitude. This yields a phase curve with anoffset 12105 but with the same width as the second case 12095 and areduced amplitude curve 12100 with the damping equivalent to that of thesecond example. If there is an offset in the phase versus frequencycurve as there is in this third example, prior art phase locked-loopelectronics will not maintain stable operation. For example, if thephase set-point was made to be 90 degrees, it would never be possible tofind a frequency in curve 12105 where this condition was met. Oneexample of these things occurring in a practical situation is in DRFTPFM when the tip crosses from an electric domain with one orientation toa second domain with another orientation. The response induced by thesecond domain will typically have a phase offset with respect to thefirst. This is, in fact where the large contrast in DFRT PFM phasesignals originates.

FIG. 9 shows the cantilever response when the applied potential is sweptfrom dc to 2 MHz using the DFRT PFM apparatus. Three resonance peaks arevisible. Depending on the cantilever probe and the details of thetip-sample contact mechanics, the number, magnitude, breadth andfrequency of the peaks is subject to change. Sweeps such as these areuseful in choosing the operating points for imaging and othermeasurements. In a practical experiment, any or all of these resonancepeaks or the frequencies in between could be exploited by the methodssuggested above.

FIGS. 17-19 shows a measurement that can be made using DFRT PFMtechniques. A phase image 19010 shows ferroelectric domains written ontoa sol-gel PZT surface. Because of the excellent separation betweentopography and PFM response possible with DFRT PFM, the phase imageshows only piezo response, there is no topographic roughness couplinginto the phase. The written domains appear as bright regions. Thewriting was accomplished by locally performing and measuring hysteresisloops by applying a DC bias to the tip during normal DFRT PFM operation.This allows the local switching fields to be measured. The piezo phase19030 during a measurement made at location 19020 and the amplitude19040 are plotted as a function of the applied DC bias voltage. Theloops were made following Stephen Jesse et al, Rev. Sci. Inst. 77,073702 (2006). Other loops were taken at the bright locations in image19010, but are not shown in the Figure.

DFRT PFM is very stable over time in contrast to single frequencytechniques. This allows time dependent processes to be studied as isdemonstrated by the sequence of images, 19010, 19050, 19060, 19070 and19080 taken over the span of 1.5 hours. In these images, the writtendomains are clearly shrinking over time.

In AC mode atomic force microscopy, relatively tiny tip-sampleinteractions can cause the motion of a cantilever probe oscillating atresonance to change, and with it the resonant frequency, phase andamplitude and deflection of the probe. Those changes of course are thebasis of the inferences that make AC mode so useful. With contactresonance techniques the contact between the tip and the sample also cancause the resonant frequency, phase and amplitude of the cantileverprobe to change dramatically.

The resonant frequency of the cantilever probe using contact resonancetechniques depends on the properties of the contact, particularly thecontact stiffness. Contact stiffness in turn is a function of the localmechanical properties of the tip and sample and the contact area. Ingeneral, all other mechanical properties being equal, increasing thecontact stiffness by increasing the contact area, will increase theresonant frequency of the oscillating cantilever probe. Thisinterdependence of the resonant properties of the oscillating cantileverprobe and the contact area represents a significant shortcoming ofcontact resonance techniques. It results in “topographical crosstalk”that leads to significant interpretational issues. For example, it isdifficult to know whether or not a phase or amplitude change of theprobe is due to some sample property of interest or simply to a changein the contact area.

The apparatus used in contact resonance techniques can also cause theresonant frequency, phase and amplitude of the cantilever probe tochange unpredictably. Examples are discussed by Rabe et al., Rev. Sci.Instr. 67, 3281 (1996) and others since then. One of the most difficultissues is that the means for holding the sample and the cantilever probeinvolve mechanical devices with complicated, frequency dependentresponses. Since these devices have their own resonances and damping,which are only rarely associated with the sample and tip interaction,they may cause artifacts in the data produced by the apparatus. Forexample, phase and amplitude shifts caused by the spurious instrumentalresonances may freely mix with the resonance and amplitude shifts thatoriginate with tip-sample interactions.

It is advantageous to track more than two resonant frequencies as theprobe scans over the surface when using contact resonance techniques.Increasing the number of frequencies tracked provides more informationand makes it possible to over-constrain the determination of variousphysical properties. As is well known in the art, this is advantageoussince multiple measurements will allow better determination of parametervalues and provide an estimation of errors.

Since the phase of the cantilever response is not a well behavedquantity for feedback purposes in PFM, we have developed other methodsfor measuring and/or tracking shifts in the resonant frequency of theprobe. One method is based on making amplitude measurements at more thanone frequency, both of which are at or near a resonant frequency. FIG.13 illustrates the idea. The original resonant frequency curve 14010 hasamplitudes A₁ 14030 and A₂ 14020, respectively, at the two drivefrequencies f₁ and f₂. However, if the resonant frequency shifted to alower value, the curve shifts to 14050 and the amplitudes at themeasurement frequencies change, A′₁ 14035 increasing and A′₂ 14025decreasing. If the resonant frequency were higher, the situation wouldreverse, that is the amplitude A′₁ at drive frequency f₁ would decreaseand A′₂ at f₂ would increase.

There are many methods to track the resonant frequency with informationon the response at more than one frequency. One method with DFRT PFM isto define an error signal that is the difference between the amplitudeat f₁ and the amplitude at f₂, that is A₁ minus A₂. A simpler examplewould be to run the feedback loop such that A₁ minus A₂=0, althoughother values could equally well be chosen. Alternatively both f₁ and f₂could be adjusted so that the error signal, the difference in theamplitudes, is maintained. The average of these frequencies (or evensimply one of them) provides the user with a measure of the contactresonance frequency and therefore the local contact stiffness. It isalso possible to measure the damping and drive with the two values ofamplitude. When the resonant frequency has been tracked properly, thepeak amplitude is directly related to the amplitude on either side ofresonance. One convenient way to monitor this is to simply look at thesum of the two amplitudes. This provides a better signal to noisemeasurement than does only one of the amplitude measurements. Other,more complicated feedback loops could also be used to track the resonantfrequency. Examples include more complex functions of the measuredamplitudes, phases (or equivalently, the in-phase and quadraturecomponents), cantilever deflection or lateral and/or torsional motion.

The values of the two amplitudes also allow conclusions to be drawnabout damping and drive amplitudes. For example, in the case of constantdamping, an increase in the sum of the two amplitudes indicates anincrease in the drive amplitude while the difference indicates a shiftin the resonant frequency.

Finally, it is possible to modulate the drive amplitude and/orfrequencies and/or phases of one or more of the frequencies. Theresponse is used to decode the resonant frequency and, optionally,adjust it to follow changes induced by the tip-sample interactions.

FIG. 10 shows the results of a measurement of a piezo-electric materialusing DFRT PFM methods. Contact mode is used to image the sampletopography 10010 and contact resonance techniques used to image thefirst frequency amplitude 10020, the second frequency amplitude 10030,the first frequency phase 10040 and the second frequency phase 10050. Inthis experiment, the two frequencies were chosen to be close to thefirst contact resonance, at roughly the half-maximum point, with thefirst frequency on the lower side of the resonance curve and the secondon the upper side. This arrangement allowed some of the effects ofcrosstalk to be examined and potentially eliminated in subsequentimaging.

Another multiple frequency technique is depicted in FIG. 2, an apparatusfor using the present invention with a conductive cantilever, and themethods for its use may also be advantageous in examining the effects ofcrosstalk with a view to potentially eliminating them in subsequentimaging. For this purpose the inventors refer which to this apparatusand method as Dual Frequency Piezo Force Microscopy (DF PFM). In the DFPFM apparatus of FIG. 2 the response to driving the tip voltage of thecantilever, due to the piezoelectric action acting through the contactmechanics, will typically change as the probe is scanned over thesurface. The first signal will then be representative of changes in thecontact mechanics between the tip and sample. The second signal willdepend both on contact mechanics and on the piezo electrical forcesinduced by the second excitation signal between the tip and sample.Differences between the response to the first excitation and theresponse to the second are thus indicative of piezoelectric propertiesof the sample and allow the contact mechanics to be separated from suchproperties.

As noted, the user often does not have independent knowledge about thedrive or damping in contact resonance. Furthermore, models may be oflimited help because they too require information not readily available.In the simple harmonic oscillator model for example, the drive amplitudeA_(drive), drive phase Φ_(drive), resonant frequency ω₀ and qualityfactor Q (representative of the damping) will all vary as a function ofthe lateral tip position over the sample and may also vary in timedepending on cantilever mounting schemes or other instrumental factors.In conventional PFM, only two time averaged quantities are measured, theamplitude and the phase of the cantilever (or equivalently, the in-phaseand quadrature components). However, in dual or multiple frequencyexcitations, more measurements may be made, and this will allowadditional parameters to be extracted. In the context of the SHO model,by measuring the response at two frequencies at or near a particularresonance, it is possible to extract four model parameters. When the twofrequencies are on either side of resonance, as in the case of DFRT PFMfor example, the difference in the amplitudes provides a measure of theresonant frequency, the sum of the amplitudes provides a measure of thedrive amplitude and damping of the tip-sample interaction (or qualityfactor), the difference in the phase values provides a measure of thequality factor and the sum of the phases provides a measure of thetip-sample drive phase.

Simply put, with measurements at two different frequencies, we measurefour time averaged quantities, A₁, A₂, Φ₁, Φ₂ that allow us to solve forthe four unknown parameters A_(drive), Φ_(drive), ω₀ and Q.

FIG. 15-16 illustrates the usefulness of measuring the phase as a meansof separating changes in the quality factor Q from changes in the driveamplitude A_(drive). Curve 18010 shows the amplitude response of anoscillator with a resonance frequency of ω₀=320 kHz, a quality factorQ=110 and a drive amplitude A_(drive)=0.06 nm. Using DFRT PFMtechniques, the amplitude A₁ 18012 is measured at a drive frequency f₁and the amplitude A₂ 18014 is measured at a drive frequency f₂. Curve18030 shows what happens when the Q value increases to 150. The firstamplitude A₁ 18032 increases because of this increase in Q, as does thesecond amplitude A2 18034. Curve 18050 shows what happens when thequality factor Q, remains at 110 and the drive amplitude A_(drive)increases from 0.06 nm to 0.09 nm. Now, the amplitude measurements madeat f1 18052 and f2 18054 are exactly the same as in the case where the Qvalue increased to 150, 18032 and 18034, respectively. The amplituderesponse does not separate the difference between increasing the Q valueor increasing the drive amplitude A_(drive).

This difficulty is surmounted by measuring the phase. Curves 18020,18040 and 18060 are the phase curves corresponding to the amplitudecurves 18010, 18030 and 18050 respectively. As with the amplitudemeasurements, the phase is measured at discrete frequency values, f₁ andf₂. The phase curve 18020 remains unchanged 18060 when the driveamplitude A_(drive) increases from 0.06 nm to 0.09 nm. Note that thephase measurements 18022 and 18062 at f₁ for the curves with the samequality factor are the same, as are the phase measurements 18024 and18064 at f₂. When the quality factor Q increases, the f₁ phase 18042decreases and the f₂ phase 18044 increases. These changes clearlyseparate drive amplitude changes from Q value changes.

In the case where the phase baseline does not change, it is possible toobtain the Q value from one of the phase measurements. However, as inthe case of PFM and thermal modulated microscopy, the phase baseline maywell change. In this case, it is advantageous to look at the difference18070 in the two phase values. When the Q increases, this difference18080 will also increase.

If we increase the number of frequencies beyond two, other parameterscan be evaluated such as the linearity of the response or the validityof the simple harmonic oscillator model

Once the amplitude, phase, quadrature or in-phase component is measuredat more than one frequency, there are numerous deductions that can bemade about the mechanical response of the cantilever to various forces.These deductions can be made based around a model, such as the simpleharmonic oscillator model or extended, continuous models of thecantilever or other sensor. The deductions can also be made using apurely phenomenological approach. One simple example in measuringpassive mechanical properties is that an overall change in theintegrated amplitude of the sensor response implies a change in thedamping of the sensor. In contrast, a shift in the “center” of theamplitude in amplitude versus frequency measurements implies that theconservative interactions between the sensor and the sample havechanged.

This idea can be extended to more and more frequencies for a betterestimate of the resonant behavior. It will be apparent to those skilledin the art that this represents one manner of providing a spectrum ofthe sensor response over a certain frequency range. The spectralanalysis can be either scalar or vector. This analysis has the advantagethat the speed of these measurements is quite high with respect to otherfrequency dependent excitations.

In measuring the frequency response of a sensor, it is not required toexcite the sensor with a constant, continuous signal. Other alternativessuch as so-called band excitation, pulsed excitations and others couldbe used. The only requirement is that the appropriate reference signalbe supplied to the detection means.

FIG. 14 shows one embodiment of a multi-frequency approach, with eightfrequencies being driven f₁ through f₈. As the resonance curve changesin response to tip-surface interactions, a more complete map of thefrequency response is traced out. This may be particularly useful whenmeasuring non-linear interactions between the tip and the sample becausein that case the simple harmonic oscillator model no longer applies. Theamplitude and phase characteristics of the sensor may be significantlymore complex. As an example of this sort of measurement, one can drivethe cantilever at one or more frequencies near resonance and measure theresponse at nearby frequencies.

Scanning ion conductance microscopy, scanning electrochemicalmicroscopy, scanning tunneling microscopy, scanning spreading resistancemicroscopy and current sensitive atomic force microscopy are allexamples of localized transport measurements that make use ofalternating signals, sometimes with an applied dc bias. Electrical forcemicroscopy, Kelvin probe microscopy and scanning capacitance microscopyare other examples of measurement modes that make use of alternatingsignals, sometimes with an applied dc bias. These and other techniquesknown in the art can benefit greatly from excitation at more than onefrequency. Furthermore, it can also be beneficial if excitation of amechanical parameter at one or more frequencies is combined withelectrical excitation at the same or other frequencies. The responsesdue to these various excitations can also be used in feedback loops, asis the case with Kelvin force microscopy where there is typically afeedback loop operating between a mechanical parameter of the cantileverdynamics and the tip-sample potential.

The described embodiments of the invention are only considered to bepreferred and illustrative of the inventive concept. The scope of theinvention is not to be restricted to such embodiments. Various andnumerous other arrangements may be devised by one skilled in the artwithout departing from the spirit and scope of the invention.

What is claimed is:
 1. An atomic force microscope which processes asample, comprising: an atomic force microscope cantilever which has aprobe tip which moves relative to a surface of the sample; an actuatorfor the cantilever which actuates the probe tip; a magnetic connectionto the probe tip which modulates a magnetic state of a circuit partadjacent to the probe tip; a controller, controlling said cantilever bycreating relative movement between the probe tip of the cantilever andthe sample and measuring values used to control said cantilever, saidcontroller using both first and second frequencies to cause saidrelative movement of the cantilever, where said first and secondfrequencies are different frequencies, and said controller including afirst frequency source that creates said first frequency and a secondfrequency source that creates said second frequency, said controllercoupling said first frequency to the actuator to drive actuation of theprobe tip, said controller coupling said second frequency to change themagnetic state of the circuit part adjacent to the probe tip connectionto vary a characteristic of the surface of the cantilever probe usingsaid second frequency; and said controller using said first and secondfrequencies and a measured value indicative of said relative movement,to provide information indicative of a surface of the sample.
 2. Themicroscope as in claim 1, wherein said controller includes a lock-inamplifier locking in the first and second frequencies.
 3. The microscopeas in claim 1, wherein said cantilever is controlled using bothamplitude and phase of the cantilever at both of said first and secondfrequencies.
 4. The microscope as in claim 1, wherein the measured valueincludes a value indicative of motion measured at a first eigenvalueexcited by the first frequency source, and also includes another valueindicative of surface-state dependent properties at a second eigenvalueexcited by the second frequency source.
 5. The microscope as in claim 4,wherein the measured value is based on amplitude of the cantilever. 6.The microscope as in claim 1, wherein the circuit part is a surface nearthe sample being measured.